Rational approximations to solutions of linear differential equations

Abstract

Rational approximations of Pade and Pade type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.The purpose of this paper is to present simple and complete proofs.of several results on best rational approximations to solutions of differential equations. Our methods are based on studies of particular differential invariants associated with graded subrings of Picard-Vessiot extensions of differential fields (1). The origin of these studies is the famous Kolchin's problem (2) on the extension of "Roth's theorem" to approximation of solutions of arbitrary algebraic differential equations by rational functions. The Roth's theorem is used here in the broad sense as the property of numbers or functions that was proved by Roth (3) for algebraic numbers. For numbers 6, the Roth's theorem means that for any E > 0 and arbitrary rational integers p and q, 10 -(p/q)l > IqL 2-8 for q 2 qo(E). For functions y(x), the Roth's theorem means that for any E > 0 and arbitrary poly-P(x) nomials P(x) and In this paper, we prove functional solutions of linear differential equations over k(x) and also prove Schmidt's theorem and the generalization of Roth's theorem for the simultaneous approximations in several normings. The methods of this paper are also useful in our complete solution of the Kolehin problem for arbitrary algebraic differential equations. We want to acknowledge that C. Osgood announced a year ago an effectivization of Roth's theorem for algebraic functions and recently announced the Roth's and Schmidt's theorems for solutions of linear differential equations. After the Roth theorem (3) in 1955, its p-adic and g-adic generalizations appeared (11)(12)(13). This development was summarized by Mahler in his book (11), where he presented a general "approximation theorem" on rational approximations of algebraic numbers in archimedian and nonarchimedian metrics.Let Pi, P2, . Pr+r'+ r be a fixed system of r + r' + r" distinct primes. Also, for a real number and for a p-adic number

Équations différentielles linéaires et majorations de multiplicités

“…Le présent travail tire sa source de [4] et [13]. Dans [4], G. Chudnovsky établit le théorème ci-dessus sous l'hypothèse que toutes les singularités du système (^) sont régulières.…”

A subspace theorem for ordinary linear differential equations

“…This result is best possible of its type but non-effective in the sense that it does not yield an upper bound for the largest value of q. Later C. F. Osgood [4] proved an effective "Rothtype" result for the solutions of algebraic differential equations, and D. V. Chudnovsky and G. V. Chudnovsky [1] used wronskian methods and graded subrings of Picard-Vessiot extensions to prove an effective analogue of Roth's Theorem for the solutions of ordinary linear differential equations.…”

“…The purpose of this paper is to present a proof of an effective subspace theorem for the differential function field case based on the methods used by D. V. Chudnovsky and G. V. Chudnovsky [1] in their function field analogue of Roth's Theorem.…”

On applications of diophantine approximations

“…The results on G-functions can be considerably improved with the use of methods of graded Pade approximations proposed in the functional case by D. V. and G. V. Chudnovsky (8) and applied to number theory in ref. 3.…”

“…These parametrized solutions have the form Y/X = P(N)/Q(N), A-fixed, for P(x), Q(x) E Q[x]. Parametrized solutions can be determined as exceptionally good rational approximations P(x)/Q(x) of a real branch y = y(x) of F(x, y) -0 at x = oo-and there are only finitely many such exceptionally good approximations (8). With the exception ofparametrized solutions, the Thue equation I has only finitely many integral solutions (X, Y; N)for a fixed A.…”